We prove that a strong version of Chang's Conjecture together with [Formula: see text] implies there are no [Formula: see text]-Aronszajn trees.

Define a point in a topological space to be if it is fixed by every self-homotopy of the space, i.e. every self-map of the space which is homotopic to the identity, and define a point to be if it has a neighborhood whose covering dimension is one. In this paper, we show that every Peano continuum is homotopy equivalent to a reduced form in which the one-dimensional points which are not homotopically fixed form a disjoint union of open arcs. In the case of one-dimensional Peano continua, this presents the space as a compactification of a null sequence of open arcs by the homotopically fixed subspace.

Every Peano continuum has a strong deformation retract to a continuum, that is, one with no strongly contractible subsets attached at a single point. In a deforested continuum, each point with a one-dimensional neighborhood is either fixed by every self-homotopy of the space, or has a neighborhood which is a locally finite graph. A minimal deformation retract of a continuum (if it exists) is called its . Every one-dimensional Peano continuum has a unique core, which can be obtained by deforestation. We give examples of planar Peano continua that contain no core but are deforested.

Type II topoisomerases are enzymes that change the topology of DNA by performing strand-passage. In particular, they unknot knotted DNA very efficiently. Motivated by this experimental observation, we investigate transition probabilities between knots. We use the BFACF algorithm to generate ensembles of polygons in Z(3) of fixed knot type. We introduce a novel strand-passage algorithm which generates a Markov chain in knot space. The entries of the corresponding transition probability matrix determine state-transitions in knot space and can track the evolution of different knots after repeated strand-passage events. We outline future applications of this work to DNA unknotting.